This phenomenon of regularization by noise is now well-studied in several situations. A lot of work has been devoted to the additive case \(\sigma \equiv 1\)  and for several kind of processes. One can think of the seminal work of [1]}, [2]} for strong solutions of this equation when \(w\)  is a Brownian motion. In this additive case and when \(w\)  is a Brownian motion, Davie [3]} exhibit a new and stronger notion of uniqueness and proved that the previous equation has a strong solution and enjoys "path-by-path" uniqueness whenever \(b\in L^\infty (\mathbb {R}^d)\) , whereas standard theory and counterexample require \(b\)  to be Lipschitz continuous. One can consult [4]} for a deep discussion about notions of solutions in the additive case.
